Covariance Regression Analysis

ABSTRACT

This article introduces covariance regression analysis for a p-dimensional response vector. The proposed method explores the regression relationship between the p-dimensional covariance matrix and auxiliary information. We study three types of estimators: maximum likelihood, ordinary least squares, and feasible generalized least squares estimators. Then, we demonstrate that these regression estimators are consistent and asymptotically normal. Furthermore, we obtain the high dimensional and large sample properties of the corresponding covariance matrix estimators. Simulation experiments are presented to demonstrate the performance of both regression and covariance matrix estimates. An example is analyzed from the Chinese stock market to illustrate the usefulness of the proposed covariance regression model. Supplementary materials for this article are available online.

KEYWORDS:

• Covariance matrix estimation
• Covariance regression
• Portfolio management
• Positive definiteness

Supplementary Materials

This supplementary material studies the model and parameter estimation for n ≥ 1, regardless of whether p is fixed or goes to infinity. Specifically, Section 1 introduces the covariance regression model for n ≥ 1, and Section 2 discusses technical conditions for fixed p. In addition, Section 3 presents parameter estimation and asymptotic results, and Section 4 proposes an algorithm for constrained estimation. Detailed proofs of theoretical results are given in Appendices A and B.

Acknowledgment

The authors are grateful to the editor, the AE, and the anonymous referee for their insightful comments and constructive suggestions.

Funding

Wei Lan’s research was supported by National Natural Science Foundation of China (NSFC, 11401482, 71532001). Hansheng Wang’s research was supported in part by National Natural Science Foundation of China (NSFC, 11131002, 11271031, 71532001), the Business Intelligence Research Center at Peking University, and the Center for Statistical Science at Peking University.